# Mathematics

*(For 2015–2016 academic year)*

*Professors*Hart

*,*Lantz, Robertson

*(Chair)*, Saracino

*,*Schult,

*Strand, Valente*

*Assistant Professors*Ay, Christensen, Howard, Jiménez Bolaños, Seo

*Visiting Assistant Professor*Stephens

There are many good reasons to study mathematics: preparation for a career, use in another field, or the beauty of the subject itself. Students at Colgate who major in mathematics go on to careers in medicine, law, or business administration as well as areas of industry and education having an orientation in science. Non-majors often require mathematical skills to carry on work in other disciplines, and all students can use the study of mathematics to assist them in forming habits of precise expression, in developing their ability to reason logically, and in learning how to deal with abstract concepts. There are also many people who view mathematics as an art form, to be studied for its own intrinsic beauty.

All mathematics courses are open to qualified students. Entering first-year students who have successfully completed at least three years of secondary school mathematics, including trigonometry, should be adequately prepared for

**MATH 111.**Students who have studied calculus in secondary school are often ready to enter

**MATH 112**or

**113.**

**(or equivalent calculus experience approved by the department) is required for admission to a major or minor in mathematics.**

## Major Program

MATH 111, 112The requirements for a major in mathematics are as follows:

**MATH 113**and**214**completed by the end of the sophomore year.**MATH 250**completed by the end of the sophomore year with a grade of C or better.**MATH 320**and**323**- Five additional mathematics courses numbered 300 or above. One of these must be a senior research experience which will normally be
**MATH 399**or a 400-level course. With department chair pre-approval, on occasion the research experience may be satisfied by an independent study. See “Advanced Placement and Transfer Credit” below for additional information. - Majors who are planning to undertake graduate study in mathematics are advised to take
**MATH 421**and**424.**

**The requirements for a**

## Minor Program in Mathematics

**minor are as follows:**

**MATH 113**and**214.****MATH 250**completed with a grade of C or better.- Either
**MATH 320**or**323.** - Two other mathematics courses numbered

**The requirements for a minor are as follows:**

## Minor Program in Applied Mathematics

**MATH 113**and**214.****MATH 250**completed with a grade of C– or better.**MATH 308.**- Two of the following courses:
**MATH 307, 310, 311, 312, 313, 315, 316, 317, 329, MATH/BIOL 302, MATH 458**(formerly**MATH 407), MATH/BIOL 481.**See “Advanced Placement and Transfer Credit” below for additional information.

**In order to graduate**with a major in mathematics, a minor in mathematics, or a minor in applied mathematics, the student must have a GPA of at least 2.00 in mathematics courses counted for the major or minor.

The department also strongly recommends that students pursuing a major or a minor in mathematics complete

**COSC 101**or its equivalent.

**To qualify for honors in mathematics, majors must take, as one of the courses required for the major, a course at the 400 level**

## Honors and High Honors

**.**Majors must have a GPA of at least 3.30 in mathematics courses counted for the major. For high honors, the corresponding GPA must be at least 3.70. Candidates for honors must also perform satisfactorily on the honors examination, which is given once each semester and covers

**MATH 320**and

**323.**

Based upon the result of the honors examination, a student may be invited to stand for high honors. A candidate for high honors must, under the guidance of a faculty member of the department, write a high honors paper during the senior year and make an oral presentation of the results. In order for high honors to be awarded, the department must accept this paper and presentation as being of high honors quality. The high honors candidate may register for an independent study course so that the paper satisfies the senior experience requirement.

**See “Honors and Awards: Mathematics” in Chapter VI.**

## Awards

**Students should review the**

## Calculus Placement

**MATH 111, 112,**and

**113**course descriptions for information on topics and prerequisites, or consult with a department faculty member. In general, students are encouraged to enroll in a higher-level course. Students may drop back from

**MATH 112**to

**MATH 111**within the first three weeks, subject to available space in an acceptable

**MATH 111**section.

**Students who have taken the Calculus-BC, Calculus-AB, or Statistics Advanced Placement exam of the College Entrance Examination Board will be granted credit according to the following policy:**

## Advanced Placement and Transfer Credit

- Students earning 4 or 5 on the Calculus-BC Advanced Placement exam will receive credit for
**MATH 111**and**112.**Students earning 3 on the Calculus BC exam will receive credit only for**MATH 111**. - Students earning 4 or 5 on the Calculus-AB Advanced Placement exam will receive credit for
**MATH 111.** - Students earning 4 or 5 on the Statistics Advanced Placement exam will receive credit for
**MATH 105.** - There are no other circumstances under which a student will receive credit at Colgate for a mathematics course taken in high school.

**MATH 250, 320,**or

**323,**but must pass these courses

*at Colgate*and must take them as regularly scheduled courses, not as independent studies. At most, two transfer or independent studies courses may be counted toward a major or minor.

**The Department of Educational Studies offers a teacher education program for majors in mathematics who are interested in pursuing a career in elementary or secondary school teaching. Please refer to “Educational Studies.”**

## Teacher Certification

## Related Majors

- Computer Science

- Economics
- Mathematical Systems Biology

**Colgate sponsors several study-abroad programs that can support continued work toward a major in mathematics. These include, but are not limited to, the Wales Study Group (U.K.), the Australia Study Group, the Australia II Study Group, and the Manchester Study Group (U.K.). For more information about these programs, see “Off-Campus Study” in Chapter VI.**

## Study Groups

## Course Offerings

*MATH courses count toward the Natural Sciences and Mathematics area of inquiry requirement, unless otherwise noted.*

**101 Precalculus Mathematics**

*A. Strand*

A study of the following types of functions, their properties, and their graphs: polynomials, rational functions, trigonometric, exponential, and logarithmic functions. This 0.25-credit course is intended for students whose background in mathematics may be deficient. Its objective is to lay a foundation for the study of calculus and concurrent enrollment is

**MATH 111**is required. Credit is contingent upon completion (regardless of grade) of MATH 111. Prerequisite: permission of the instructor. Offered in the fall only.

**105 Introduction to Statistics**

*Staff*

An introduction to the basic concepts of statistics. Topics include experimental design, descriptive statistics, correlation, regression, basic probability, mean tendencies, the central limit theorem, point estimation with errors, hypothesis testing for means, proportions, paired data, and the chi-squared test for independence. Emphasis is on statistical reasoning rather than computation, although computation is done via spreadsheet. Prerequisite: three years of secondary school mathematics. Note: This course is not open to students who have either received credit for or are currently enrolled in

**CORE 143S,**the former

**MATH 102,**or

**MATH 317.**

**111 Calculus I**

*Staff*

An introduction to the basic concepts of differential and integral calculus including limits and continuity; differentiation of algebraic, trigonometric, exponential, and logarithmic functions; applications of the derivative to curve sketching, related rates, and maximum-minimum problems; Riemann sums and the definite integral; and the fundamental theorem of calculus. Prerequisite: three years of secondary school mathematics including trigonometry.

**112 Calculus II**

*Staff*

A continuation of the study of calculus begun in

**MATH 111,**including the calculus of inverse trigonometric functions, techniques of integration, improper integrals, l’Hôpital’s rule and indeterminate forms, applications of integration, and Taylor series. Prerequisite:

**MATH 111**with a grade of C– or higher or equivalent experience in a secondary school calculus course.

**113 Multivariable Calculus**

*Staff*

The calculus of functions of two or three variables. Among the topics considered are surfaces in three-dimensional space, partial derivatives, maxima and minima, and multiple integrals. Prerequisite:

**MATH 111**with a grade of C– or higher or equivalent experience in a secondary school calculus course.

**214 Linear Algebra**

*A. Ay, D. Lantz, D. Schult, A. Strand*

A study of systems of linear equations, matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and diagonalization. Prerequisite:

**MATH 113**or co-registration in

**MATH 113.**

**250 Number Theory and Mathematical Reasoning**

*E. Hart, D. Howard, D. Saracino, K. Valente*

Questions about the positive integers 1, 2, 3 . . . have fascinated people for thousands of years. The ancient Greeks noted the existence of right triangles with sides of integral length, corresponding to equations such as 32 + 42 = 52 and 52 + 122 = 132. Is there a way of describing all such “Pythagorean Triples”? As another example, we observe that 5 = 12 + 22, 13 = 22 + 32, 17 = 12 + 42, while none of the primes 7, 11, or 19 can be expressed as the sum of two squares. Is there a pattern? Does it continue forever? This course focuses on such equations as a means for introducing students to the spirit and methods of modern mathematics. The emphasis throughout is on developing the ability to construct logically sound mathematical arguments and communicate these arguments in writing. Prerequisite:

**MATH 112**or

**113**with a grade of C or

**better, or permission of instructor.**

**302 Systems Biology**

This course is crosslisted as

**BIOL 302**. For course description, see “Biology: Course Offerings.”

**307 Dynamical Systems and Chaos**

*D. Schult*

A dynamical system is a set of mathematical rules which specify change over time. Examples include models of the solar system or of population levels. The goal of this course is to introduce some of the concepts and discoveries that have been made over the past fifty years about discrete dynamical systems. Repeated application of non-linear functions can lead to complex phenomena. Topics include fixed points, stability, chaos, symbolic dynamics, fractals, Lyapunov exponents, the Julia set, and the Mandelbrot set. Prerequisite:

**MATH 214.**Offered in the fall only, in alternate years.

**308 Differential Equations**

*A. Ay, S. Jiménez Bolaños, D. Schult, G. Seo, A. Strand*

A study of ordinary differential equations with associated initial condition. First order equations, linear second order equations with constant coefficients, systems of differential equations, other selected topics, and applications. Prerequisite:

**MATH 112**and

**MATH 113**or permission of instructor.

**310 Combinatorial Problem Solving**

*D. Howard, A. Robertson*

This course develops methods to solve combinatorial (finite) problems arising in mathematics, computer science, and other areas from the natural and social sciences. Enumeration and graph theory are the main subjects. Topics include recurrence relations, generating functions, inclusion-exclusion, modeling with graphs, trees and searching, graph coloring, and network algorithms. The emphasis is on problem solving rather than theory. Prerequisite:

**MATH 112**or permission of instructor. Offered in the spring only, in alternate years.

**311 Partial Differential Equations**

*A. Ay, S. Jiménez Bolaños, D. Schult, G. Seo, A. Strand*

This course explores mathematics as it is applied to the physical sciences. Mathematical topics may include boundary value problems, partial differential equations, special functions, Fourier series and transforms, Green’s functions, and approximate solution methods. Prerequisite:

**MATH 308**or permission of instructor

**.**Offered in the spring only, in alternate years.

**312 Applied Mathematics: Social Sciences**

*D. Schult, G. Seo*

How do we translate problems from the world into solvable mathematical problems? Mathematical modeling is the art of creating mathematical problems whose solutions are useful for real world problems. Methods such as scaling, qualitative analysis, limits of predictability, and simple random models are discussed. Applications considered arise from economics, political science, and sociology. Prerequisite:

**MATH 214**or permission of instructor. Offered in the spring only.

**313 Functions of a Complex Variable**

*J. Christensen, D. Lantz, D. Schult*

An introductory study of functions in the complex plane. Topics include complex numbers and functions, the theory of differentiation and integration of complex functions, sequences and series of complex functions, conformal mapping. Special attention is given to Cauchy’s integral theorem. Prerequisite:

**MATH 112**and

**113.**Offered in the spring only, in alternate years.

**315 Mathematical Biology**

*A. Ay, D. Schult, G. Seo*

This course provides an introduction to the use of continuous and discrete mathematical models in the biological sciences. Biological topics may include single and multispecies population dynamics, modeling of infectious diseases, regulation of cell function, molecular interactions, neural and biological oscillators, ecology, cancer biology, and virus dynamics. Mathematical techniques include modeling, phase plane analysis, bifurcation diagrams, perturbation theory, and computer simulations. Prerequisite:

**MATH 112**and

**113**. Offered in the fall only, in alternate years.

**316 Probability**

*E. Hart, A. Robertson*

An introduction to the basic concepts of discrete and continuous probability: axioms and properties of probability, standard counting techniques, conditional probability, important random variables and their discrete and continuous distributions, expectation, variance, and joint distribution functions. Additional topics may include: Poisson processes, Markov chains, and Monte Carlo methods. Prerequisite:

**MATH 112**and

**113**(or co-registration of

**113**) or permission of instructor. Offered in the fall.

**317 Mathematical Statistics**

*E. Hart, A. Robertson*

The standard methods in statistics are developed with mathematical rigor. Topics include parameter estimation, Bayesian estimation, the Central Limit Theorem, hypothesis testing, regression, analysis of variance, moment generating functions, and nonparametric statistics. Applications of these tools are studied, with the choice of topics determined by the instructor. Prerequisite:

**MATH**

**316.**Offered in the spring only, in alternate years.

**320 Abstract Algebra I**

*E. Hart, D. Lantz, D. Saracino, K. Valente*

An introduction to the basic structures of abstract algebra including groups, rings, integral domains, and fields. Prerequisite:

**MATH 250**with a grade of C or better

**.**Offered in the spring.

**323 Real Analysis I**

*J. Christensen, A. Robertson, D. Saracino*

A rigorous treatment of the basic concepts of real analysis, including limits, continuity, the derivative, and the Riemann integral. Prerequisites:

**MATH 112, 113**and

**250**with a grade of C or better

**.**Offered in the fall.

**327 Geometry**

*D. Lantz*

A study of several geometrical systems, with emphasis upon a development of Euclidean geometry that meets current standards of rigor. Prerequisite:

**MATH 250.**Offered in the fall only, in alternate years.

**329 Numerical Analysis**

*A. Ay, D. Schult, G. Seo*

An introductory treatment of methods used for numerical approximation. Topics include roots of equations, simultaneous linear equations, quadrature, and other fundamental processes using high speed computing devices. Prerequisite:

**MATH 113.**Offered in the fall only, in alternate years.

**331 Number Theory II**

*D. Saracino*

This course continues the study of number theory begun in

**MATH 250**and includes the Quadratic Reciprocity Law of Gauss, Diophantine equations, and topics from algebraic number theory. Prerequisite:

**MATH 320**or permission of instructor. Offered in the fall only, in alternate years.

**342 Topology**

*J. Christensen, E. Hart*

An introduction to both point-set topology and basic algebraic topology. Topics include metric spaces, topological spaces, compactness, connectedness, the classification of surfaces, mod-2 homology, and the Jordan curve theorem. Additional topics that demonstrate connections with analysis, dynamics, and algebra are determined by the instructor based on student interest. Prerequisites:

**MATH 250**with a grade of C or better

**.**Offered in the spring only, in alternate years.

**380/480 Special Topics in Mathematics**

*Staff*

The topic for this course varies depending on the needs and backgrounds of students and interests of the instructor. Students should consult the instructor for the specific content of the course and prerequisites.

**399 Mathematical Problem Solving**

*D. Howard, D. Lantz, A. Robertson*

This capstone seminar presents students with numerous and varied problems, drawn from many different mathematical areas, both pure and applied. There are weekly problem sets in addition to the presentation of a semester-long “project problem.” Fulfills the senior experience for the major, the course is open only to seniors.

**421 Abstract Algebra II**

*D. Lantz, D. Saracino, K. Valente*

A careful and intensive study of topics such as group theory, ring theory, field theory, and Galois theory. Prerequisite:

**MATH 320**with a grade of B or better or permission of instructor

**.**Offered in the fall only, in alternate years.

**424 Real Analysis II**

*A. Ay, J. Christensen, A. Robertson*

Topics for this course are selected from among the following: metric spaces, sequences and series of functions, the Lebesgue integral. Prerequisite:

**MATH 323**with a grade of B or better or permission of instructor

**.**Offered in the spring only, in alternate years.

**452 Mathematical Logic**

*D. Saracino*

This course deals with one or more topics in mathematical logic, chosen from among the following: naive and axiomatic set theory, propositional and predicate calculus, completeness and compactness theorems, first-order model theory, recursive functions, and Gödel’s Incompleteness Theorem. Prerequisite:

**MATH 320**with a grade of B or better and permission of instructor. Offered in the fall only, in alternate years.

**458 Real-time Nonlinear Dynamics and Chaos**

*D. Schult*

An introduction to the techniques and concepts used to analyze real-time dynamic models that involve nonlinear terms. Applications are emphasized and demonstrate the universality of chaotic solution behavior. This course is team-taught by members of the physics and mathematics departments. Students should enroll through the department for which they intend to use the credit. (Formerly

**MATH 407**.) Prerequisites:

**MATH 308**or

**PHYS 431**(formerly

**PHYS 302)**. Offered in the spring only, in alternate years. This course is crosslisted as

**PHYS 458**(formerly

**PHYS 407**).

**481 Modeling of Biological Systems**

This course is crosslisted as

**BIOL 481**. For course description, see “Biology: Course Offerings.”

**291, 391, 491 Independent Study**

*Staff*

Open to qualified students with permission of department chair.